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dc.contributor.advisorSantosh S. Vempala and Daniel A. Spielman.en_US
dc.contributor.authorDeshpande, Amit Jayanten_US
dc.contributor.otherMassachusetts Institute of Technology. Dept. of Mathematics.en_US
dc.date.accessioned2007-09-28T13:13:54Z
dc.date.available2007-09-28T13:13:54Z
dc.date.copyright2007en_US
dc.date.issued2007en_US
dc.identifier.urihttp://hdl.handle.net/1721.1/38935
dc.descriptionThesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2007.en_US
dc.descriptionIncludes bibliographical references (p. 51-52).en_US
dc.description.abstractCan one compute a low-dimensional representation of any given data by looking only at its small sample, chosen cleverly on the fly? Motivated by the above question, we consider the problem of low-rank matrix approximation: given a matrix A..., one wants to compute a rank-k matrix (where k << min{m, n}) nearest to A in the Frobenius norm (also known as the Hilbert-Schmidt norm). We prove that using a sample of roughly O(k/[epsilon]) rows of A one can compute, with high probability, a (1 + [epsilon])-approximation to the nearest rank-k matrix. This gives an algorithm for low-rank approximation with an improved error guarantee (compared to the additive [epsilon]... guarantee known earlier from the work of Frieze, Kannan, and Vempala) and running time O(Mk/[epsilon]), where M is the number of non-zero entries of A. The proof is based on two sampling techniques called adaptive sampling and volume sampling, and some linear algebraic tools. Low-rank matrix approximation under the Frobenius norm is equivalent to the problem of finding a low-dimensional subspace that minimizes the sum of squared distances to given points. The general subspace approximation problem asks one to find a low-dimensional subspace that minimizes the sum of p-th powers of distances (for p > 1) to given points. We generalize our sampling techniques and prove similar sampling-based dimension reduction results for subspace approximation. However, the proof is geometric.en_US
dc.description.statementofresponsibilityby Amit Jayant Deshpande.en_US
dc.format.extent52 p.en_US
dc.language.isoengen_US
dc.publisherMassachusetts Institute of Technologyen_US
dc.rightsM.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.en_US
dc.rights.urihttp://dspace.mit.edu/handle/1721.1/7582
dc.subjectMathematics.en_US
dc.titleSampling-based algorithms for dimension reductionen_US
dc.typeThesisen_US
dc.description.degreePh.D.en_US
dc.contributor.departmentMassachusetts Institute of Technology. Department of Mathematics
dc.identifier.oclc166267550en_US


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